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  "body": "\n## Example\n\nLet $\\alpha$ be a real number.\nFor each natural number $k$, let\n$$\\binom{\\alpha}{k} = \\frac{\\alpha (\\alpha - 1) \\cdots (\\alpha - k + 1)}{k!}.$$\nThe binomial series is the [[power series|0VP56B4EMX]] $\\sum_{k=0}^{\\infty} \\binom{\\alpha}{k} x^k$.\nFor $|x| \u003c 1$ its sum is $(1 + x)^{\\alpha}$.\n\n## Remarks\n\nIf $\\alpha$ is a natural number, then $\\binom{\\alpha}{k} = 0$ for $k \u003e \\alpha$. The series is then the binomial formula for $(1 + x)^{\\alpha}$.\nOtherwise the [[radius of convergence is 1|0KX1MXJ5MR]].\nFor $\\alpha = -1$ it is the geometric series $\\sum (-1)^k x^k = 1/(1 + x)$.\nFor $\\alpha = 1/2$ it gives $\\sqrt{1 + x} = 1 + x/2 - x^2/8 + \\cdots$.\n\n## Explanations\n\n### L1 human\n\nThe binomial series extends the binomial formula to exponents that are not natural numbers.\nThe expansion then has infinitely many terms, and it holds only for small $x$.\n",
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